My colleague Scott Steketee recently shared a problem that makes for a nice exploration with Web Sketchpad.
Using the websketch below (and here), construct two segments, AB and CD, and their midpoints. Now, connect the endpoints to form quadrilateral ACBD. By dragging A, B, C, and D what types of special quadrilaterals (kite, parallelogram, rhombus, etc.) can you make, and what can be said about the diagonals of the quadrilateral in each case?
When experimenting with the diagonals, I found myself wishing that the midpoint of one diagonal was attached to the other diagonal, or that the two midpoints remained coincident now matter how I dragged the quadrilateral’s vertices. On page 2 of the websketch, there are some additional tools so that you can construct ABCD to have these properties.
The video here demonstrates these constructions.